Sunday, February 08, 2009

Deep Thinking About Math

(I)s mathematics a human tool, or is reality, in some fundamental way, mathematics?

IDEAS: Can you imagine an alternate universe in which we invent a different type of math? What might that look like?

LIVIO: Let me start with this silly idea - the isolated jellyfish. Imagine that all the intelligence resided not in humans, but in some isolated jellyfish at the bottom of the Pacific Ocean. This jellyfish - all it would feel would be the pressure of the water, the temperature of the water, the motion of the water. Would this jellyfish have invented the natural numbers - 1, 2, 3, 4, 5, and so on? I think probably not, because there's nothing to count there - everything this creature would have felt would have been continuous rather than discrete, so this creature might have invented a completely different type of mathematics.

IDEAS: So, instead of jellyfish math we have math that reflects our abilities?

LIVIO: Why did the ancient Babylonians and Greeks and so on start with arithmetic and geometry? I think that largely this is because of our particular perception system. We are very, very good at seeing edges of things; we can very well tell what is an object, what is the background of the object; we can tell individual objects very well, we can also tell very well whether a line is straight or not, just with our eye.

IDEAS: So, at some level, mathematics is a human invention?

LIVIO: The mathematics we use to explain the universe were not actually chosen arbitrarily. Imagine that you do this very simple experiment where you put pebbles in a jar. You put first three pebbles, and five more pebbles, and you want the mathematical tool that will predict to you how many pebbles are in the jar. You think this is idiotic - it's mathematical addition - three plus five. If you do the same experiment with drops of oil, and you say three drops and five drops, you just have a pool of oil, the same thing does not apply to the second example.

Humans at some level chose the mathematical tools based on them being suitable for the particular problem.